Topology, Geometry, and Group Theory

小行星9803374 大约50年前，Aleksandrov，Busemann和Wald引入了定义比黎曼流形更一般的度量空间的“曲率”的上下界的想法。 亚历山德罗夫主要对非负曲率的情况及其在欧氏三维空间中凸多面体曲面上的应用感兴趣。 兴趣在这一主题是重新约十年前，当格罗莫夫指出，拓扑和群论的原因，非积极弯曲的情况下，更有趣的是积极弯曲的情况下，主要原因是，普遍覆盖的非积极弯曲的空间是收缩。 格罗莫夫还指出，有许多多面体的例子，这样的空间。 特别是，山雀建筑就是这样的例子。 在过去的几年里，教授查尼和戴维斯合作研究非正弯曲多面体和应用群论。 他们主要感兴趣的团体是Coxeter团体和Artin团体。 这两种类型的组都与反射生成的组相关联。 查尼和戴维斯在这些领域也完成了单独的研究。 他们将继续对这些领域的研究，以及对密切相关的主题，如山雀建筑物和自同构群的属性。 由反射产生的群在数学和自然界的许多地方都有出现（例如，在晶体学中）。 欧几里得平面的规则镶嵌的对称群就是这样的群。 布泽尔的绘图显示了非欧几里德平面几何中规则平铺的例子。 这些也与反射组相关联。 查尼和戴维斯教授对更抽象的例子感兴趣。 事实证明，抽象反射群（考克斯特群）总是可以被实现为非正弯曲空间的对称群，就像欧几里得平面和非欧几里得平面的情况一样。 ***

9803374 Davis About 50 years ago, Aleksandrov, Busemann and Wald introduced the idea of defining upper and lower bounds for the ``curvature'' of more general metric spaces than Riemannian manifolds. Aleksandrov was interested primarily in the case of nonnegative curvature and its applications to convex polyhedral surfaces in Euclidean 3-space. Interest in this subject was renewed about ten years ago when Gromov pointed out that for topological and group theoretic reasons the nonpositively curved case was much more interesting than the positively curved case, the main reason being that the universal cover of a nonpositively curved space is contractible. Gromov also pointed out that there were many polyhedral examples of such spaces. In particular, Tits buildings are such examples. For the past several years, Professors Charney and Davis have collaborated in investigating nonpositively curved polyhedra and applications to group theory. The groups in which they are primarily interested are Coxeter groups and Artin groups. Both types of groups are associated to groups generated by reflections. Both Charney and Davis have also accomplished separate research in these areas. They will continue their research on these areas as well as on closely related topics such as the properties of Tits buildings and their automorphism groups. Groups generated by reflections occur in many places in mathematics and in nature (for example, in crystallogrophy). The symmetry groups of regular tilings of the Euclidean plane are such groups. Escher's drawings show examples of regular tilings in non-Euclidean plane geometry. These are also associated with reflection groups. Professors Charney and Davis are interested in such examples in a more abstract setting. It turns out that abstract reflection groups (Coxeter groups) can always be realized as the symmetry groups of a space that is nonpositively curved, much as in the case of the Euclidean and non-Euclidean planes. ***